Just as a marble added to the cup must sit on top of all others, an electron entering a metal cannot be understood without reference to all the pre-existing electrons. It interacts with them, but it also becomes quantum-mechanically entangled with them. From this perspective, it is remarkable that this second class of emergent electron – the elementary electron plus the interactions and entanglement with all others in a metal – resembles an elementary electron at all. In fact, it does, but this idea (called Fermi liquid theory) took many decades to understand.

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To understand this third class, this time consider a magnetic crystal. I will describe a particular type of magnetic crystal, called a quantum spin ice, although there are other examples showing similar effects. The magnetism originates from individual atoms, each of which has its own quantum magnetic field called a ‘spin’. (Strictly, the atoms are electrically charged ions, but this is not relevant to the story.) Starting from these spins, we can once again blur our eyes until we see only a quantum field. What form do the emergent particles take?

Three types emerge. The first is an emergent electron. This is deeply surprising, since this time there were no elementary electrons present. There were only ions with magnetic fields. As Anderson said, the whole is not just more than the sum of the parts: it is also different. The second emergent particle, the magnetic monopole, is stranger still. Whereas an electron has an electric charge, the monopole has a ‘magnetic charge’. It looks like the north pole of a magnet without the south. We learn in school that this is impossible: cutting a magnet in two gives two smaller magnets, each with a north and south pole, not separate north and south poles. But as emergent particles within quantum spin ices, magnetic monopoles can exist. The third type of particle is an emergent photon. Like the elementary photon, it sets the maximum speed at which signals can propagate within the crystal. It also mediates interactions between the emergent electrons and monopoles.

All three of these emergent particles are truly remarkable since they feature properties that are not present at the level from which they emerge. Emergent particles in other crystals are similarly able to take forms never seen among elementary particles.